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63
Affine Matching With Bounded Sensor Error: A Study of Geometric Hashing and Alignment
, 1991
"... Affine transformations of the plane have been used in a number of modelbased recognition systems, in order to approximate the effects of perspective projection. The mathematics underlying these methods is for exact data, where there is no positional uncertainty in the measurement of feature points ..."
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Cited by 47 (5 self)
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Affine transformations of the plane have been used in a number of modelbased recognition systems, in order to approximate the effects of perspective projection. The mathematics underlying these methods is for exact data, where there is no positional uncertainty in the measurement of feature points. In practice, various heuristics are used to adapt the methods to real data with uncertainty. In this paper, we provide a precise analysis of affine point matching under uncertainty. We obtain an expression for the range of affineinvariant values that are consistent with a given set of four points, where each data point lies in a disk of radius e. This analysis reveals that the range of affineinvarint values depends on the actual xypositions of the data points. That is, when there is uncertainty in the data then the representation is no longer invariant with respect to the Cartesian coordinate system. This is problematic for the geometric hashing method, because it means that the precomputed lookup table used by that method is not correct when there is positional uncertainty in the sensor data. We analyze the effect that this has on the probability that the geometric hashing method will find false positive matches of a model to an image, and contrast this with a similar analysis of the alignment method.
Solving the quintic by iteration
 Acta Math
, 1989
"... Equations that can be solved using iterated rational maps are characterized: an equation is ‘computable ’ if and only if its Galois group is within A5 of solvable. We give explicitly a new solution to the quintic polynomial, in which the transcendental inversion of the icosahedral map (due to Hermit ..."
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Cited by 15 (5 self)
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Equations that can be solved using iterated rational maps are characterized: an equation is ‘computable ’ if and only if its Galois group is within A5 of solvable. We give explicitly a new solution to the quintic polynomial, in which the transcendental inversion of the icosahedral map (due to Hermite and Kronecker) is replaced by a purely iterative algorithm. The algorithm requires a rational map with icosahedral symmetries; we show all rational maps with given symmetries can be described using the classical theory of invariant polynomials. 1 Introduction. According to Dickson, Euler believed every algebraic equation was solvable by radicals [2]. The quadratic formula was know to the Babylonians; solutions of cubic and quartic polynomials by radicals were given by Scipione del Ferro, Tartaglia, Cardano and Ferrari in the mid1500s. Abel’s proof of the insolvability of the general quintic
The geometry of continued fractions and the topology of surface singularities
, 2005
"... We survey the use of continued fraction expansions in the algebraical and topological study of complex analytic singularities. We also prove new results, firstly concerning a geometric duality with respect to a lattice between plane supplementary cones and secondly concerning the existence of a cano ..."
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Cited by 12 (1 self)
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We survey the use of continued fraction expansions in the algebraical and topological study of complex analytic singularities. We also prove new results, firstly concerning a geometric duality with respect to a lattice between plane supplementary cones and secondly concerning the existence of a canonical plumbing structure on the abstract boundaries (also called links) of normal surface singularities. The duality between supplementary cones gives in particular a geometric interpretation of a duality discovered by Hirzebruch between the continued fraction expansions of two numbers λ> 1 and λ/λ−1.
An Approach to Image Retrieval for Image Databases
 Lecture Notes in Computer Science
, 1993
"... In this paper, a method is discussed to store and retrieve images efficiently from an image database on the basis of the data structure called E() representation. The E() representation is a spatial knowledge representation preserving the spatial information between objects embedded in symbolic imag ..."
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Cited by 11 (0 self)
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In this paper, a method is discussed to store and retrieve images efficiently from an image database on the basis of the data structure called E() representation. The E() representation is a spatial knowledge representation preserving the spatial information between objects embedded in symbolic images as an iconic index for the purpose of efficient image retrieval. The image retrieval method is invariant under, at least, the affine transformation (i.e. translation, rotation and scale) and is able to deal with substantial object occlusion. A metric is defined to express similarity between symbolic images. Initial experiments carried out for two applications show that the image retrieval method is very efficient and robust to similarity retrieval in image databases. Together with the inherent high parallelism, it makes the method a promising image retrieval method. keywords: image database, image indexing, similarity retrieval, spatial relations, E() representation, metric, spatial quer...
A Probabilistic Approach to Geometric Hashing using Line Features
 Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition
, 1996
"... Most current object recognition algorithms assume reliable image segmentation, which in practice is often not available. We examine the combination of the Hough Transform with a variation of Geometric Hashing as a technique for modelbased object recognition in seriously degraded single intensity im ..."
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Most current object recognition algorithms assume reliable image segmentation, which in practice is often not available. We examine the combination of the Hough Transform with a variation of Geometric Hashing as a technique for modelbased object recognition in seriously degraded single intensity images. Prior work on the performance analysis of geometric hashing has focused on point features and shown its noise sensitivity. This paper uses line features to compute recognition invariants in a potentially more robust way. We investigate the statistical behavior of these line features analytically. Various viewing transformations, which 2D (or flat 3D) objects undergo during image formation, are considered. For the case of affine transformations, which are often suitable substitutes for more general perspective transformations, we show experimentally that the technique is noise resistant and can be used in highly occluded environments. 1 1 Introduction Visual object recognition can ...
Minimal information to determine affine shape equivalence
, 2000
"... Participants judged the affine equivalence of 2 simultaneously presented 4point patterns. Performance level (d') varied between 1.5 and 2.7, depending on the information available for solving the correspondence problem (insufficient in Experiment la, superfluous in Experiment lb, and minimal i ..."
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Cited by 7 (1 self)
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Participants judged the affine equivalence of 2 simultaneously presented 4point patterns. Performance level (d') varied between 1.5 and 2.7, depending on the information available for solving the correspondence problem (insufficient in Experiment la, superfluous in Experiment lb, and minimal in Experiments lc, 2a, 2b) and on the exposure time (unlimited in Experiments 1 and 2a and 500 ms in Experiment 2b), but it did not vary much with the complexity of the affine transformation (rotation and slant in Experiment 1 and same plus tilt in Experiment 2). Performance in Experiment 3 was lower with 3point patterns than with 4point patterns, whereas blocking the trials according to the affine transformation parameters had little effect. Determining affine shape equivalence with minimalinformation displays is based on a fast assessment of qualitatively or quasiinvariant properties such as convexity/ concavity, parallelism, and collinearity. When an object is viewed from different positions, its projected shape on the retina varies, yet its perceived shape often remains the same. A door, for example, looks rectangular even though its projected shape is generally trapezoidal.
Mathematics, mathematicians, and mathematics education
 Bulletin Amer. Math. Society
, 2005
"... I am one of a growing number of research mathematicians who are substantially engaged with school mathematics education. Such outreach has a long and honorable tradition. In this lecture, I illustrate some of the ways that I think this can be helpful, and even essential. Upon his retirement in 1990 ..."
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I am one of a growing number of research mathematicians who are substantially engaged with school mathematics education. Such outreach has a long and honorable tradition. In this lecture, I illustrate some of the ways that I think this can be helpful, and even essential. Upon his retirement in 1990 as president of the ICMI, 1 Jean Pierre Kahane spoke perceptively of the intimate connection between mathematics and mathematics education in the following terms: • In no other living science is the part of presentation, of the transformation of disciplinary knowledge to knowledge as it is to be taught (transformation didactique) so important at a research level. • In no other discipline, however, is the distance between the taught and the new so large. • In no other science has teaching and learning such social importance. • In no other science is there such an old tradition of scientists ’ commitment to educational questions.
Forceassembly with friction
 IEEE Transactions on Robotics and Automation
, 1994
"... If an admittance control law is properly designed, a workpiece can be guided into a fixture using only the contact forces for guidance (forceassembly). Previously, we have shown that: 1) a space of accommodation control laws that will ensure forceassembly without friction always exists, and 2) as ..."
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Cited by 5 (1 self)
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If an admittance control law is properly designed, a workpiece can be guided into a fixture using only the contact forces for guidance (forceassembly). Previously, we have shown that: 1) a space of accommodation control laws that will ensure forceassembly without friction always exists, and 2) as friction is increased, a control law that allows forceassembly can be obtained as long as the forces associated with positional misalignment are characteristic. A single accommodation control law that allows forceassembly at the maximum value of friction can be obtained by an optimization procedure. The single accommodation control law obtained by the optimization procedure, however, is not unique. There exists a space of accommodation control laws that will allow forceassembly at, or below, the value of friction that marginally violates the characteristic forces condition. Here, for the purpose of the accommodation control law design, a set of linear sufficient conditions is used to generate accommodation basis matrices. Any nonnegative linear combination of the accommodation basis matrices that, when combined, yields a positive definite accommodation matrix is guaranteed to provide forceassembly at or below a specified value of friction. (Basis
The decline and rise of geometry in 20th century North America
, 1999
"... While I will begin with my own evidence for the decline of geometry in this century and my own description on how such a decline has proceeded, my basic theme is hopeful. Geometry has not died because it is essential to many other human activities and because it is so deeply embodied in how humans t ..."
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Cited by 5 (0 self)
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While I will begin with my own evidence for the decline of geometry in this century and my own description on how such a decline has proceeded, my basic theme is hopeful. Geometry has not died because it is essential to many other human activities and because it is so deeply embodied in how humans think. With the introduction of computers with rich graphical capacities and the
Map transformation of geographic space
, 1961
"... Many geographic and economic models of human behavior in a spatial context indicate that the measuring rod of the geodesist or surveyor is less relevant than a scaling of distance in temporal or monetary units. It is necessary to take into account not only the shape of the earth but also the realiti ..."
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Cited by 4 (1 self)
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Many geographic and economic models of human behavior in a spatial context indicate that the measuring rod of the geodesist or surveyor is less relevant than a scaling of distance in temporal or monetary units. It is necessary to take into account not only the shape of the earth but also the realities of transportation on this surface. Automobiles, trains, airplanes, and other media of transport can be considered to have the effect of modifying distances measured in temporal or monetary units in a complicated manner. Different distance relations, however, can be interpreted as different types of geometry. A geographically natural approach is to attempt to map this geometry to a plane, in a manner similar to the preparation of maps of the terrestrial surface. If the transformations were dependent only on a factor of proportionality, as in the conversion from miles to kilometers, there would be no difficulty. The facts are considerably more complex. The geometry with which we must deal is rarely Euclidean, and it is, in general, impossible to prepare a plane map preserving all distance relations, just as isometric transformations of a spherical surface to a plane are not possible. Maps preserving distance relations from one point are easily achieved, however, whatever the units of measurement.